Modeling of lophotrichous bacteria reveals key factors for swimming reorientation

Lophotrichous bacteria swim through fluid by rotating their flagellar bundle extended collectively from one pole of the cell body. Cells experience modes of motility such as push, pull, and wrapping, accompanied by pauses of motor rotation in between. We present a mathematical model of a lophotrichous bacterium and investigate the hydrodynamic interaction of cells to understand their swimming mechanism. We classify the swimming modes which vary depending on the bending modulus of the hook and the magnitude of applied torques on the motor. Given the hook’s bending modulus, we find that there exist corresponding critical thresholds of the magnitude of applied torques that separate wrapping from pull in CW motor rotation, and overwhirling from push in CCW motor rotation, respectively. We also investigate reoriented directions of cells in three-dimensional perspectives as the cell experiences different series of swimming modes. Our simulations show that the transition from a wrapping mode to a push mode and pauses in between are key factors to determine a new path and that the reoriented direction depends upon the start time and duration of the pauses. It is also shown that the wrapping mode may help a cell to escape from the region where the cell is trapped near a wall.

: Time evolutions of flagellar rotation rate (top), cell body rotation rate (middle) and swimming speed (bottom) for two overwhirling cases when the applied torque is given as τ = 0.0015 gµm 2 /s 2 (blue) and 0.004 gµm 2 /s 2 (red). For the small value of τ = 0.0015, the overall swimming direction is backward (negative speed); however, for the large value of τ = 0.004, the cell swims forward (positive speed). Figure S2: Average swimming speeds and motor rotation rates as functions of the applied torque generated by the flagellar motor. The bending modulus of the filament is fixed as a = 0.003 gµm 3 /s 2 , and the bending modulus of the hook is set as a hook = a 35 (top row) and a hook = a 15 (bottom row). The direction of motor rotation is CCW for left column (a, c) and CW for right column (b, d), where the positive (negative) values of τ lead to CCW (CW) rotation. The motor frequency denoted by f is defined to be positive when the motor turns CCW and vice versa. The forward swimming speed V f is defined as the velocity of the cell body in the direction from the motor point to the other pole of the cell body. This figure shows, in general, that average swimming speeds and motor rotation rates increase linearly with the increasing applied torque for each mode of motility. Figure S3: Swimming modes as functions of the bending modulus of the hook (a hook ) and the applied torque (τ ) when the bending modulus of the filament is fixed at a = 0.002 gµm 3 /s 2 (top row), 0.0025 gµm 3 /s 2 (middle row) and 0.0035 gµm 3 /s 2 (bottom row). In each row, the motor rotates either CCW (left panel) or CW (right panel). Different shapes of markers represent the stable motion of the push (■), overwhirling (⋆), pull ( • ), and wrapping (▲) modes. The markers (−) indicate that the bacterium does not show either wrapping or pull mode for a given simulation time, because the hook is too flexible and the applied torque is close to the critical value. Colors indicate the average swimming speeds measured for each mode of motility after the simulation reaches the stable steady motion. For different values of the bending modulus of the flagellar filament, the swimming patterns are similar to each other. However, thresholds that separate push from overwhirling and pull from wrapping are shifted upward as the bending modulus of the filament increases. Figure S4: Rotation rates of the motor and the cell body (top) and swimming speed (bottom) of a cell over a sequence of pull, wrapping, pause, and push modes. Each mode is derived by implementing the applied torque as in Fig. 4(a) of the main text. Positive and negative values of flagellar rotation rates correspond to CCW and CW rotation, respectively. Moreover, the cell body counterrotates to the flagellum. If the swimming speed (V f ) is positive, the cell swims forward, and if it is negative the cell swims backward.   Fluid viscosity (µ) 0.01 × 10 −4 g(µm s) −1 Parameter for repulsive force (C) 2 × 10 3 gs −2 Minimum distance allowed between the filament and the cell body (D) 0.1 µm Regularization parameter (ε) 3∆s Translational drag coefficient (α 1 ) 1.212 × 10 −7 g(µm s) −1 Translational drag coefficient (α 2 ) 4 × 10 −6 gs −1 Rotational drag coefficient (β) 1.212 × 10 −7 gµm s −1 Time step (∆t) 2 × 10 −8 s  Video S1: Swimming modes of a polarly-flagellated bacterium whose helical filament is intrinsically left-handed. Each row shows two stable motions when the motor rotates either CCW (top panel) or CW (bottom panel). These two stable motions are separated by the thresholds of applied torque, τ ccw or τ cw , respectively. When the applied torque |τ | is below |τ ccw | or |τ cw |, a cell exhibits push or pull mode, respectively (left panel). When |τ | is above |τ ccw | or |τ cw |, a cell undergoes overwhirling or wrapping mode, respectively (right panel).
Video S2: A series of swimming modes in the order of pull, wrapping, pause, and push modes. The colored path is a trajectory of the centroid of the cell body (left panel) as the cell consecutively experiences swimming modes in accordance with the applied torque of the motor in time (right top panel). The corresponding swimming speed over time is shown in the right bottom panel.
Video S3: Effect of a pause on cells' reorientation when cells go through wrapping-pause-push modes. The left panel displays the cases with four different pause initiation times, P k I = (0.15 + 0.0017k) s for k = 0, 3, 6, 9, while the pause duration is fixed as P 7 D = 0.06 s. The right panel shows the cases with four different pause durations, P j D = (0.025 + 0.005j) s for j = 0, 1, 4, 7, while the pause initiation time is fixed as P 0 I = 0.15 s. Four different simulations in each panel are shown simultaneously, while all cells take the same path until 0.15 s and they change their trajectories depending on the pause duration or pause initiation time.
Video S4: Swimming trajectories of cells going through a wrapping mode near the wall. We consider four cells with respect to four combinations of two different initial cell rotation angles (180 • , 337.5 • ) and two different initial heights (h 0 = 1.0, 2.5 µm), and place them parallel to the wall. Four different simulations are simultaneously displayed when τ = −0.004 gµm 2 /s 2 is set for all times.
Video S5: Escaping cells from the wall by switching swimming mode to a wrapping mode. When a cell settles down and is trapped near the wall during a pull mode (left panel) or a push mode (right panel), switching to a wrapping mode helps the cell to escape from the wall.